Abstract
The sequence of Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, …, defined by F0 = 0, F1 = 1, F n+2 = F n + F n+1, played an important role in the solution of one of the Hilbert Problems. The Fibonacci sequence was used in 1970 by the Russian mathematician Y.V. Matijasevič to solve the Tenth Problem of Hilbert. The Tenth Problem of Hilbert was the problem of existence of an algorithm for deciding solvability of Diophantine equations. Matijasevič [8] [9] made use of divisibility properties of the Fibonacci sequence to prove that every recursively enumerable set is Diophantine. This solved Hilbert’s Tenth Problem in the negative.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Hoggatt, V. E. Jr. “Some More Fibonacci Diophantine Equations.” The Fibonacci Quarterly Vol. 9 (1971): pp. 437 and 448.
Jones, J.P. “Diophantine Representation of the Set of Fibonacci Numbers.” The Fibonacci Quarterly, Vol. 13 (1975): pp. 84–88. MR 52 3035.
Jones, J.P. “Diophantine Representation of the Set of Lucas Numbers.” The Fibonacci Quarterly, Vol. 14 (1976): p. 134. MR 53 2818.
Jones, J.P., Sato, D., Wada, H. and Wiens, D. “Diophantine Representation of the Set of Prime Numbers.” American Mathematical Monthly, Vol. 83 (1976), pp. 449–464. MR 54 2615.
Kiss, P. “Diophantine Representation of Generalized Fibonacci Numbers.” Elementa der Mathematik, Birkhauser Verlag, Basel, Switzerland, Vol. 34–6 (1979): pp. 129–132, MR81g 10021.
Kiss, P. and Varnai, F. “On Generalized Pell Numbers.” Math. Seminar Notes (Kobe University, Japan), Vol. 6 (1978): pp. 259–267.
Lucas, E. Nouv. Corresp. Math., Vol. 2 (1876): pp. 201–206.
Matijasevič, Y.V. “Enumerable Sets are Diophantine.” Doklady Akademii Nauk, Vol. 191 (1970): pp 279–282. English translation: Soviet Math. Doklady Vol. (1970): pp. 354–358. MR 41 3390.
Matijasevič, Y.V. “Diophantine Representation of Enumerable Predicates.” Izvestija Akademii Nauk SSSR, Serija Matematič eskaya, Vol. 35 (1971): pp. 3–30. Mathematics of the USSR-Izvestija, Vol. 5 (1971): pp. 1–28. MR 43 54.
Putnam, H. “An Unsolvable Problem in Number Theory.” Jour. Symbolic Logic Vol. 25 (1960): pp. 220–232.
Vorobsev, N.N. Fibonacci Numbers, Moscow, Nauka, 1984. ( Russian).
Wasteels, J. Mathesis (3), 2 (1902), pp. 60–62. Cf. L.E. Dickson, History of the Theory of Numbers, Vol. I, 1919, Chelsea Publishing Co., N.Y. p. 405.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Kluwer Academic Publishers
About this chapter
Cite this chapter
Jones, J.P. (1990). Diophantine Representation of Fibonacci Numbers Over Natural Numbers. In: Bergum, G.E., Philippou, A.N., Horadam, A.F. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1910-5_22
Download citation
DOI: https://doi.org/10.1007/978-94-009-1910-5_22
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-7352-3
Online ISBN: 978-94-009-1910-5
eBook Packages: Springer Book Archive